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December  2011, 31(4): 1325-1346. doi: 10.3934/dcds.2011.31.1325

Fine properties of minimizers of mechanical Lagrangians with Sobolev potentials

Alessio Figalli 1, and  Vito Mandorino 2,

1. 

University of Texas at Austin, Department of Mathematics, 2515 Speedway Stop C1200, Austin, TX 78712-1202, United States
2. 

Université Paris-Dauphine, CEREMADE UMR CNRS 7534, Place du Maréchal de Lattre de Tassigny, 75775 Paris Cedex 16, France

Received  November 2009 Revised  August 2010 Published  September 2011

In this paper we study the properties of curves minimizing mechanical Lagrangianwhere the potential is Sobolev. Since a Sobolev functionis only defined almost everywhere, no pointwise results can be obtained in this framework,and our point of view is shifted from single curves to measures in the space of paths.This study is motived by the goal of understanding the properties ofvariational solutions to the incompressible Euler equations.
Keywords: Non-smooth Lagrangians, action-minimizing measures, Euler-Lagrange equations, value function..
Mathematics Subject Classification: Primary: 49Kxx, 49Lxx; Secondary: 70H03, 37J5.
Citation: Alessio Figalli, Vito Mandorino. Fine properties of minimizers of mechanical Lagrangians with Sobolev potentials. Discrete & Continuous Dynamical Systems, 2011, 31 (4) : 1325-1346. doi: 10.3934/dcds.2011.31.1325
References:
[1]

L. Ambrosio and A. Figalli, Geodesics in the space of measure-preserving maps and plans, Arch. Ration. Mech. Anal., 194 (2009), 421-462. doi: 10.1007/s00205-008-0189-2.  Google Scholar

[2]

L. Ambrosio and A. Figalli, On the regularity of the pressure field of Brenier's weak solutions to incompressible Euler equations, Calc. Var. Partial Differential Equations, 31 (2008), 497-509.  Google Scholar

[3]

L. Ambrosio, N. Fusco and D. Pallara, "Functions of Bounded Variation and Free Discontinuity Problems," Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 2000.  Google Scholar

[4]

V. Arnold, Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits (French), Ann. Inst. Fourier (Grenoble), 16 (1966), 319-361. doi: 10.5802/aif.233.  Google Scholar

[5]

Y. Brenier, The least action principle and the related concept of generalized flows for incompressible perfect fluids, J. Amer. Mat. Soc., 2 (1989), 225-255. doi: 10.1090/S0894-0347-1989-0969419-8.  Google Scholar

[6]

P. Cannarsa and C. Sinestrari, "Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control," Progress in Nonlinear Differential Equations and their Applications, 58, Birkhäuser Boston, Inc., Boston, MA, 2004.  Google Scholar

[7]

C. Dellacherie and P.-A. Meyer, "Probabilities and Potential," North-Holland Mathematics Studies, 29, North-Holland Publishing Co., Amsterdam-New York, 1978.  Google Scholar

[8]

D. G. Ebin and J. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid, Annals of Math. (2), 92 (1970), 102-163. doi: 10.2307/1970699.  Google Scholar

[9]

P. Mattila, "Geometry of Sets and Measures in Euclidean Spaces. Fractals and Rectifiability," Cambridge Studies in Advanced Mathematics, 44, Cambridge University Press, Cambridge, 1995.  Google Scholar

[10]

A. I. Shnirel'man, The geometry of the group of diffeomorphisms and the dynamics of an ideal incompressible fluid (Russian), Mat. Sb. (N.S.), 128 (1985), 82-109. Google Scholar

[11]

A. I. Shnirel'man, Generalized fluid flows, their approximation and applications, Geom. Funct. Anal., 4 (1994), 586-620. doi: 10.1007/BF01896409.  Google Scholar

show all references

References:
[1]

L. Ambrosio and A. Figalli, Geodesics in the space of measure-preserving maps and plans, Arch. Ration. Mech. Anal., 194 (2009), 421-462. doi: 10.1007/s00205-008-0189-2.  Google Scholar

[2]

L. Ambrosio and A. Figalli, On the regularity of the pressure field of Brenier's weak solutions to incompressible Euler equations, Calc. Var. Partial Differential Equations, 31 (2008), 497-509.  Google Scholar

[3]

L. Ambrosio, N. Fusco and D. Pallara, "Functions of Bounded Variation and Free Discontinuity Problems," Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 2000.  Google Scholar

[4]

V. Arnold, Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits (French), Ann. Inst. Fourier (Grenoble), 16 (1966), 319-361. doi: 10.5802/aif.233.  Google Scholar

[5]

Y. Brenier, The least action principle and the related concept of generalized flows for incompressible perfect fluids, J. Amer. Mat. Soc., 2 (1989), 225-255. doi: 10.1090/S0894-0347-1989-0969419-8.  Google Scholar

[6]

P. Cannarsa and C. Sinestrari, "Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control," Progress in Nonlinear Differential Equations and their Applications, 58, Birkhäuser Boston, Inc., Boston, MA, 2004.  Google Scholar

[7]

C. Dellacherie and P.-A. Meyer, "Probabilities and Potential," North-Holland Mathematics Studies, 29, North-Holland Publishing Co., Amsterdam-New York, 1978.  Google Scholar

[8]

D. G. Ebin and J. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid, Annals of Math. (2), 92 (1970), 102-163. doi: 10.2307/1970699.  Google Scholar

[9]

P. Mattila, "Geometry of Sets and Measures in Euclidean Spaces. Fractals and Rectifiability," Cambridge Studies in Advanced Mathematics, 44, Cambridge University Press, Cambridge, 1995.  Google Scholar

[10]

A. I. Shnirel'man, The geometry of the group of diffeomorphisms and the dynamics of an ideal incompressible fluid (Russian), Mat. Sb. (N.S.), 128 (1985), 82-109. Google Scholar

[11]

A. I. Shnirel'man, Generalized fluid flows, their approximation and applications, Geom. Funct. Anal., 4 (1994), 586-620. doi: 10.1007/BF01896409.  Google Scholar

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